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If something does a complete rotation per second, everyone agrees it has an angular frequency of 2π radians per second. It is well-known that one complete oscillation is equal to the angle of 2π radians, so the article as written isn't necessarily wrong as 1 oscillation per second is equivalent to 2π radians per second (just different units). However, it could possibly be clearer:

Angular frequency specifies the rate of angular change and is equal in value to 2π radians times the number of oscillations per unit of time.